Probability because that rolling 2 dice through the six sided dotssuch as 1, 2, 3, 4, 5 and also 6 dots in each die.

You are watching: Probability of rolling two dice and getting a sum of 6 When 2 dice room thrown simultaneously, thus number of event can be 62 = 36 due to the fact that each die has 1 come 6 number ~ above its faces. Then the possible outcomes are presented in the listed below table. Note:

(i) The outcomes (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6) are called doublets.

(ii) The pair (1, 2) and (2, 1) are different outcomes.

Worked-out problems involving probability for rolling 2 dice:

1. two dice are rolled. Let A, B, C be the occasions of gaining a sum of 2, a amount of 3 and also a sum of 4 respectively. Then, show that

(i) A is a simple event

(ii) B and C room compound events

(iii) A and B room mutually exclusive

Solution:

Clearly, we haveA = (1, 1), B = (1, 2), (2, 1) and C = (1, 3), (3, 1), (2, 2).

(i) since A consists of a solitary sample point, that is a an easy event.

(ii) since both B and C contain an ext than one sample point, each one of them is a compound event.

(iii) since A ∩ B = ∅, A and B are mutually exclusive.

2. 2 dice are rolled. A is the occasion that the amount of the numbers shown on the two dice is 5, and also B is the occasion that at the very least one the the dice reflects up a 3. Are the two occasions (i) support exclusive, (ii) exhaustive? Give debates in assistance of her answer.

Solution:

When two dice are rolled, we have actually n(S) = (6 × 6) = 36.

Now, A = (1, 4), (2, 3), (4, 1), (3, 2), and

B = (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1,3), (2, 3), (4, 3), (5, 3), (6, 3)

(i) A ∩ B = (2, 3), (3, 2) ≠ ∅.

Hence, A and also B are not mutually exclusive.

(ii) Also, A ∪ B ≠ S.

Therefore, A and B space not exhaustive events.

More instances related to the inquiries on the probabilities because that throwing 2 dice.

3. 2 dice are thrown simultaneously. Uncover the probability of:

(i) acquiring six as a product

(ii) gaining sum ≤ 3

(iii) getting sum ≤ 10

(iv) acquiring a doublet

(v) obtaining a amount of 8

(vi) getting sum divisible through 5

(vii) acquiring sum the atleast 11

(viii) gaining a multiple of 3 as the sum

(ix) getting a total of atleast 10

(x) acquiring an even number as the sum

(xi) getting a element number together the sum

(xii) acquiring a doublet of also numbers

(xiii) getting a multiple of 2 on one die and also a many of 3 on the other die

Solution:

Two different dice space thrown concurrently being number 1, 2, 3, 4, 5 and 6 on their faces. We understand that in a solitary thrown that two different dice, the total number of possible outcomes is (6 × 6) = 36.

(i) obtaining six together a product:

Let E1 = occasion of getting six as a product. The number who product is 6 will be E1 = <(1, 6), (2, 3), (3, 2), (6, 1)> = 4

Therefore, probability ofgetting ‘six together a product’

number of favorable outcomesP(E1) = Total variety of possible outcome = 4/36 = 1/9

(ii) getting sum ≤ 3:

Let E2 = event of obtaining sum ≤ 3. The number whose amount ≤ 3 will certainly be E2 = <(1, 1), (1, 2), (2, 1)> = 3

Therefore, probability ofgetting ‘sum ≤ 3’

variety of favorable outcomesP(E2) = Total variety of possible outcome = 3/36 = 1/12

(iii) obtaining sum ≤ 10:

Let E3 = event of gaining sum ≤ 10. The number whose sum ≤ 10 will be E3 =

<(1,1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2,6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3,6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4,6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),

(6, 1), (6, 2), (6, 3), (6, 4)> = 33

Therefore, probability ofgetting ‘sum ≤ 10’

number of favorable outcomesP(E3) = Total variety of possible outcome = 33/36 = 11/12(iv)getting a doublet:Let E4 = occasion of gaining a doublet. The number which doublet will be E4 = <(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)> = 6

Therefore, probability ofgetting ‘a doublet’

number of favorable outcomesP(E4) = Total variety of possible result = 6/36 = 1/6

(v)getting a amount of 8:

Let E5 = event of acquiring a sum of 8. The number which is a amount of 8 will be E5 = <(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)> = 5

Therefore, probability ofgetting ‘a sum of 8’

number of favorable outcomesP(E5) = Total variety of possible result = 5/36

(vi)getting sum divisible through 5:

Let E6 = event of obtaining sum divisible through 5. The number whose sum divisible by 5 will certainly be E6 = <(1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4)> = 7

Therefore, probability ofgetting ‘sum divisible through 5’

number of favorable outcomesP(E6) = Total number of possible result = 7/36

(vii)getting amount of atleast 11:

Let E7 = occasion of getting sum the atleast 11. The occasions of the sum of atleast 11 will be E7 = <(5, 6), (6, 5), (6, 6)> = 3

Therefore, probability ofgetting ‘sum the atleast 11’

variety of favorable outcomesP(E7) = Total number of possible result = 3/36 = 1/12

(viii) obtaining amultiple the 3 together the sum:

Let E8 = occasion of obtaining a many of 3 as the sum. The events of a multiple of 3 together the sum will be E8 = <(1, 2), (1, 5), (2, 1), (2, 4), (3, 3), (3, 6), (4, 2), (4, 5), (5, 1), (5, 4), (6, 3) (6, 6)> = 12

Therefore, probability ofgetting ‘a many of 3 as the sum’

number of favorable outcomesP(E8) = Total variety of possible outcome = 12/36 = 1/3

(ix) obtaining a totalof atleast 10:

Let E9 = event of obtaining a total of atleast 10. The events of a full of atleast 10 will be E9 = <(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)> = 6

Therefore, probability ofgetting ‘a total of atleast 10’

variety of favorable outcomesP(E9) = Total number of possible result = 6/36 = 1/6

(x) obtaining an evennumber together the sum:

Let E10 = occasion of acquiring an even number together the sum. The events of an also number as the sum will it is in E10 = <(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 3), (3, 1), (3, 5), (4, 4), (4, 2), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)> = 18

Therefore, probability ofgetting ‘an also number as the sum

number of favorable outcomesP(E10) = Total number of possible outcome = 18/36 = 1/2

(xi) obtaining a primenumber as the sum:

Let E11 = event of gaining a element number together the sum. The occasions of a prime number together the amount will be E11 = <(1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3), (5, 2), (5, 6), (6, 1), (6, 5)> = 15

Therefore, probability ofgetting ‘a element number as the sum’

variety of favorable outcomesP(E11) = Total number of possible outcome = 15/36 = 5/12

(xii) gaining adoublet of even numbers:

Let E12 = occasion of obtaining a double of also numbers. The occasions of a double of also numbers will be E12 = <(2, 2), (4, 4), (6, 6)> = 3

Therefore, probability ofgetting ‘a double of also numbers’

number of favorable outcomesP(E12) = Total variety of possible result = 3/36 = 1/12

(xiii) getting amultiple that 2 top top one die and also a lot of of 3 on the other die:

Let E13 = event of gaining a many of 2 ~ above one die and a many of 3 on the other die. The occasions of a many of 2 on one die and a many of 3 on the various other die will certainly be E13 = <(2, 3), (2, 6), (3, 2), (3, 4), (3, 6), (4, 3), (4, 6), (6, 2), (6, 3), (6, 4), (6, 6)> = 11

Therefore, probability ofgetting ‘a multiple of 2 on one die and a many of 3 top top the various other die’

number of favorable outcomesP(E13) = Total variety of possible result = 11/36

4. Twodice space thrown. Find (i) the odds in favour of gaining the amount 5, and also (ii) theodds against getting the amount 6.

Solution:

We understand that in a single thrown of 2 die, the total numberof feasible outcomes is (6 × 6) = 36.

Let S it is in the sample space. Then,n(S) = 36.

(i) the odds in favour of getting the amount 5:

Let E1 be the occasion of getting the amount 5. Then,E1 = (1, 4), (2, 3), (3, 2), (4, 1) ⇒ P(E1) = 4Therefore, P(E1) = n(E1)/n(S) = 4/36 = 1/9⇒ odds in favour the E1 = P(E1)/<1 – P(E1)> = (1/9)/(1 – 1/9) = 1/8.

(ii) the odds versus getting the sum 6:

Let E2 it is in the occasion of acquiring the amount 6. Then,E2 = (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) ⇒ P(E2) = 5Therefore, P(E2) = n(E2)/n(S) = 5/36 ⇒ odds versus E2 = <1 – P(E2)>/P(E2) = (1 – 5/36)/(5/36) = 31/5.

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